Experimental and theoretical study of compressive mechanical 1 behavior of red sandstone after heating-cooling treatment

: In order to investigate compressive mechanical behaviors of rock materials after 8 different heating-cooling treatments, in this paper, a series of uniaxial compressive experiments 9 are carried out on red sandstone samples after various heating temperature (from 25 ℃ to 1000 ℃ ) 10 and water cooling treatments (10 ℃ ) to obtain evolution laws of mechanical property. The 11 evolution laws of peak strength, elastic modulus, primary wave velocity and micro-structure are 12 analyzed in details. And for better reflecting compressive stress-strain behaviors of red sandstone 13 after heating-cooling treatments, based on Caputo variable-order fractional calculus, considering 14 strain correlation and constant strain loading rate, we propose a novel variable-order fractional 15 constitutive model to describe stress-strain behaviors of red sandstone samples after 16 heating-cooling treatments. The validation of proposed model is well verified and a comparative 17 study between proposed variable-order fractional constitutive model and constant-order fractional 18 constitutive model is performed to highlight the advantage of proposed model. The evolutions of 19 mechanical characteristics are revealed by presented varying-order function related to strain and 20 the influence of fitting parameters on stress-strain behaviors are also discussed for deeply 21 comprehending compressive mechanical mechanism of red sandstone after heating-cooling 22 treatments. 23


Introduction 26
With development of green energy resource, geothermal energy has been recommended as an 27 alternative clean energy with high reliability, low cost and environmentally friendliness that 28 comparing to traditional energy (Lund et al. 2005 geothermal energy, heat-carrying fluid is a significant operation and the cold water is input in rock 32 stratum and high temperature rock mass will encounter and be cooled rapidly, where hot dry rock 33 within deep rock stratum is forced to cool and hot energy is released and applied (Chandrasek temperature treatment and its rheological property were also studied and modeled (Hu et al. 2019). 49 The coupling effect of high-temperature and water-cooling on the rutting resistance of rock asphalt 50 mixture is investigated and a Bayesian approach is constructed to model the dynamic stability and 51 predict resistance of rock asphalt mixture (Ren et al. 2020). From these current researches, it can 52 be demonstrated the mechanical property of rock material is deeply affected by different high 53 temperature treatments and high temperature treatment will weaken strength, elastic modulus and 54 primary wave velocity of rock material (Zhu et al. 2020 heating-cooling treatments.

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And considering to above statements, the outline of this paper is illustrated as follows. Section 2 85 introduces a series of compressive experiments that carried out on red sandstone samples after 86 various heating-cooling treatments and the variations of elastic modulus, peak strength and 87 primary wave velocity were also analyzed in details. In section 3, a novel variable-order fractional 88 constitutive model related to strain is proposed and its applicability and validation in experimental 89 stress-strain data of red sandstone samples after heating-cooling treatments will be verified. And in 90 section 4 the sensitivity of fitting parameters is discussed and the evolution mechanisms of 91 mechanical property of treated samples are revealed. Finally, several conclusions will be drawn. The red sandstone experimental samples are selected from Liuyang mountain area of Hunan 95 province, China. Cylindrical samples were drilled from full red sandstone rock mass without 96 obvious cracks and fracture on surface, whose size are 100 mm in height and 50 mm in diameter.

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In accordance with the standard requirements of International Society of Rock Mechanics [31] , the 98 parallelism and surface fatness are controlled within ±0.05 mm and ±0.02 mm, respectively. As 99 displayed in Fig. 1 on sample, the red sandstone samples that subjected to 25, 200-10, 400-10, 600-10, 800-10 and 123 1000-10℃ are selected as analyzed examples. It can be observed in Fig. 2(a) that there is a little 124 number of initial cracks within the sample at room temperature with 25℃, which indicates initial 125 red sandstone sample has almost complete microstructure. And when the sample is heated to 200℃ 126 and cooled to target temperature with 10℃, there are small number of micro-cracks and particles 127 generated on the surface of sample where in the weak junction of aperture and it symbols the 128 effect of heating-cooling treatment has appeared. With an increase in temperature of 400 and 129 600℃, not only there are a large quantity of micro-cracks generating, but also long cracks within 130 apertures begin to connect and penetrate each other. And when the heated temperature arrives at 131 800 and 1000℃, due to the softening effect of montmorillonite in water resulted by high 132 temperature water cooling, the relatively complete structure of initial sample is divided into small 133 particle with clear cracks by the trans-granular cracks, which will induce the weaken of strength of 134 red sandstone sample. 135 with 700, 800, 900 and 1000℃, the peak strength will appear a sudden drop, which can be 147 accounted for that high temperature water cooling reduces the components of quartz and clay 148 minerals within red sandstone samples and the great ductility resulted by bonding force inside 149 sample rising down.  may be interpreted that during the initial heating-cooling process, the corresponding 155 heating-cooling treatment will reduce viscous substance and eliminate relative sliding inside rock 156 material. And with processing of heating-cooling treatment on sample, the effect of temperature 157 impaction that induced by heating-cooling will cause much developments of initial cracks within 158 rock and the rising down of peak strength will appear. The maximum peak strength is 91.430 MPa 159 that corresponding to heating temperature of 500℃ and cooling temperature of 10℃. From this 160 peak point, peak strength will decrease sharply until arriving at 40.569 MPa, which symbols for 161 this specific red sandstone, heating-cooling treatment of 500-10℃ is a critical temperature 162 condition and peak strength of red sandstone can be enhanced or weakened by heating-cooling 163 treatment. 164  and ! increases with temperature until arriving at 300℃ and then it decreases with an constant 171 rate, whose changes correspond to evolutions of compaction stages of each treated samples. 172 Meanwhile, it is well-known that the P-wave velocity is closely related to elastic modulus (Zhang 173 et al. 2017) and we can see in Fig. 6 that the evolutions of P-wave velocity under different 174 heating-cooling treatments are well agreement with variations of elastic modulus. 175 property within material is varying with time or space. Corresponding to these variations, 185 variable-order fractional calculus is presented to reflect changing mechanical behaviors. 186 In recent years, variable-order fractional calculus has obtained much achievement and there are 187 many kinds of variable-order fractional calculus with different definitions (Ross and Samko 1995; 188 Mainardi 2010). In this study, for matching continuous viscoelastic deformation between 0 and 1, 189 the variable-order fractional calculus that proposed by Coimbra (Coimbra 2003) is introduced, 190 whose order is assumed as ( ). And considering to time yields, the actual expression for 0 < 191 ( ) < 1 is defined 192 where ( ) is varying-order function related to time, stress, strain, relaxation time and so on.
And then the variable-order fractional expression of strain "($) ( ) within Eq. (4) can be 196 obtained 197 It can be demonstrated that during the uniaxial compression experiments, when vertical strain 198 loading function is set as ( ) = , the Eq. (5) can be formulated 199 where represents strain rate. In this study, is vertical loading rate, i.e., = 0.003 ⁄ . 200 Then the Eq. (6) is substituted into Eq. (4) and the stress-strain constitutive model can be 201 derived as follows: 202

Determination of varying-order function 203
In order to better employ proposed variable-order fractional constitutive model to depict 204 stress-strain relationship of rock material under different experimental conditions in next section, 205 how to determine a reasonable varying-order function within proposed model is significant. It is 206 well-known that the total deformation process of rock material can be regarded as an evolution of 207 viscoelastic plasticity. As mentioned in previous researches (Koeller 1984 behavior of material. Based on above stated, in this study, the total compressive deformation 213 process of rock material is divided as two parts, i.e., pre-peak stage and peak post stage, that is 214 shown in Fig. 7. 215 with exponential form is determined, which is related to time-varying strain, that is expressed as 218 Eq. (8). And in terms of determination of varying-order function, it can be interpreted that with an 219 consumption in full viscosity of material, the ability of resisting to deformation of material is 220 gradually weaken, this attenuation that from 1 to 0 can be characterized by classical exponential 221 function, i.e., α( ) = & "($) " & . Then considering to peak post stage of compressive deformation, 222 corresponding to fractional order α exceeding to 1, a varying-order function with same 223 exponential form is assumed, i.e., α( ) = "($) " ' . Finally, in conjunction of constitutive model (Eq. 224 (7)) and presented varying-order function (Eq. (8)), a novel variable-order fractional constitutive 225 model related to strain is formulated as Eq. (9). And in next section, its applicability and validation 226 will be verified by stress-strain experimental data of red sandstone after heating-cooling treatment. 227 where α( ) is varying-order function related to strain, + is peak strain, ! and ' represent 228 controlling strain parameters within pre-peak stage and peak post stage, respectively. 229 examples. It can be obtained in Fig. 8(a) that the experimental stress-strain data is well 239 correspondence with proposed variable-order fractional model and the constant-order fractional 240 model is inconsistent with experimental stress-strain data. There are obvious differences between 241

Model validation and comparison 230
proposed variable-order fractional model and constant-order fractional model that displayed in Fig.  242 8

(b). 243
And what need to be highlighted is evolution of varying-order function and its corresponding 244 interpretation. We can see with growth of strain, when time-varying strain is less than peak strain 245 + , varying-order function will decay from 1 to 0, which can be interpreted the full viscosity of 246 material, that characterized by varying-order function equaling to 1, will be consumed to resist  Fig. 9 Comparisons among experimental stress-strain curve, predicted curves of proposed variable-order fractional model and constant-order fractional model. And variations between and varying-order function and strain. a) Comparisons and variations of red sandstone after heating and cooling treatment of 300-10℃; b) That of red sandstone after 400-10℃; c) That of red sandstone after 500-10℃; d) That of red sandstone after 600-10℃; e) That of red sandstone after 700-10℃; f) That of red sandstone after 800-10℃; g) That of red sandstone after 900-10℃; h) That of red sandstone after 1000-10℃.  the pre-peak stage in Fig. 10(a), the increasing of ! will upgrade its peak strength and elastic 272 deformation and the variations of ! has no effect on initial compaction stage. Likewise, in Fig.  273 10(b), the alterations of ' only change stress value and have no influence on tendency of 274 stress-strain curve of peak post stage. 275 (a) (b) Fig. 11 Effect of relaxation time ! and ' on stress-strain relationships of pre-peak stage and peak post stage And considering to effect of relaxation time ! and ' on stress-strain curves, Fig. 11(a) 276

Analysis of sensitivity of parameters
presents with an increase in relaxation time ! , the influence of ! on peak strength is little, but 277 it reduces the length of compaction stage. And it is similar as phenomenon in Fig. 10(b), Fig. 11 concentrating on controlling strain parameter ! and ' within pre-peak stage and peak post 281 stage, in Fig. 12(a), the influence of controlling strain ! on stress-strain behavior is same as 282 displayed in Fig. 10(a) and Fig. 11(a). It is worth noting that the variations of ' has prominent 283 effect on stress-strain curve of peak post stage. When ' rises down, the peak strength is still 284 invariant, but decreasing rate of stress rises up gradually. And it is noticeable that the decreasing of 285 ' will keep last strength of peak post stage constant and reduce the peak strength of treated 286 sample. The decreasing rate of stress will rise down gradually with an increase in controlling 287 strain ' . In short, based on above discussions on sensitivity of parameters, it can be concluded 288 that elastic modulus and relaxation time have obvious effect on peak strength and controlling 289 strain parameter will induce peak strength and evolution characteristics of stress-strain curve 290 within peak post stage, which will provide references for interpreting deformation mechanism of 291 total stress-strain curve under compressive experiments. 292 (a) (b) Fig. 12 Effect of strain controlling parameter ! and ' on stress-strain relationships of pre-peak stage and peak post stage 4 Discussions 293 The main mineral composition of red sandstone includes clastic and clay minerals, e.g., quartz, 294 feldspar, montmorillonite and illite etc. When red sandstone is subjected to water, the clay 295 minerals within red sandstone are easy to disintegrate and soften quickly, which is the main factor 296 affecting mechanical property of red sandstone. Different high temperature treatments will cause 297 variations of thermal expansion and contraction coefficient of mineral particles within rock itself, 298 which will lead to thermal stress due to disharmony of thermal deformation. When thermal stress 299 resulted by high temperature heating exceeding to tensile yield strength of crystal particles, 300 numerous micro-cracks will be developed and the internal structure of rock mass will be destroyed 301 (Zhu et al. 2020). Subsequently, thermal rupture will be induced and affect mechanical property. 302 During the processing of rising temperature, the interlayer water, crystal water and structural water 303 within rock mass will be taken off and even decomposed and then a specific mineral composition 304 with little water or no water is formed, which will weaken physical and mechanical property of 305 rock (Zhang et al. 2015). Then when the rock after high temperature treatment is subjected to 306 water for cooling rapidly, thermal shock that resulted by rapid cooling will cause temperature 307 gradient inside rock mass, which is greater than that leaded by supplying steady heated flow. And 308 the micro-cracks surrounding crystal particles will extend and penetrate and then run though the 309 whole rock sample by rapid water cooling impaction and at last, it can be concluded the surface of 310 rock is intact and a large number of cracks and fissures have been produced inside rock mass. In 311 short, the changes of physical and chemical, e.g., microstructure, mineral composition and growth 312 of cracks, after heating-cooling treatment is the main reason for weakness and damage of 313 mechanical property of rock mass. 314 5 Conclusions 315 For studying compressive mechanical property of red sandstone after heating-cooling 316 treatments including ten kinds of high temperature, a series of compressive experiments were 317 carried out and corresponding elastic modulus, peak strength and primary wave velocity were also 318 analyzed. And in order to better describe compressive stress-strain behavior of red sandstone after 319 different heating-cooling treatments, a variable-order fractional constitutive model was proposed 320 and its applicability and validation have been verified by experimental data. Several detailed 321 conclusion are expressed as follow.

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(1) A series of compressive experiments were conducted on red sandstone samples after 323 heating-cooling treatments to investigate evolution of mechanical property. Peak strength of 324 treated red sandstone samples exhibit increasing and then decreasing with heating temperature 325 and it will arrive at maximum peak strength with heating temperature of 500℃. The evolution 326 laws of elastic modulus are same as that of p-wave velocity. Both experience short increase 327 and then decrease with an increase in heating temperature.

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(2) The compressive stress-strain behaviors exhibit various distributions and especially in 329 compaction stage, high temperature water cooling will induce the growth of compaction stage 330 of treated samples. For better depicting stress-strain behaviors, based on variable-order 331 fractional calculus, considering strain correlation and constant loading strain rate, a 332 variable-order fractional constitutive model is proposed with clear physical meaning and 333 expressions. The applicability and validation of proposed variable-order fractional constitutive 334 model have been verified by obtain experimental stress-strain data.

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(3) By comparing constant-order fractional constitutive model, the advantage of proposed 336 variable-order fractional model is highlighted and it is well agreement with experimental data. 337 Then the varying-order function related to strain is presented and the variation of 338 varying-order function is used to exhibit the evolution of mechanical property. Finally, the 339 sensitivity of fitting parameters is analyzed and the effect of elastic modulus, relaxation time 340 and controlling strain on stress-strain behaviors is illustrated. The damage mechanism of 341 heating-cooling treatment on red sandstone is discussed from the micro-structure in details.

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Acknowledgment 343 The work introduced in this paper was supported by the National Key R&D Program of China 344 (2016YFC0600901), National Natural Science Foundation of China (41572334, 11572344), and 345 Fundamental Research Funds for the Central Universities (2010YL14). 346 Declaration of competing interest 347 The authors declare that they have no known competing financial interests or personal 348 relationships that could have appeared to influence the work reported in this paper. Cai W, Wang P, Fan J (2020) A variable-order fractional model of tensile and shear behaviors for 353